Alternative models NEW

In order to discern which model predictions are specific to a rational meta-memory model, we show predictions of two alternative models. The “Random” model randomly samples fixation durations from the empirical distribution. We also A more sophisticated “Random Commitment” model was developed to account for the empirical fact that last fixations are considerably longer than all other fixations. This model samples the total number of fixations on each trial from the empirical distribution and then samples the duration of each from the empirical distribution of non-final fixations. If a target is not recalled before reaching the final fixation, the model continues to fixate on the current cue until the target is recalled or the 15 second time limit is reached.

Measuring memory strength NEW

The model predicts that people will spend more time looking at the cue for which the memory of the corresponding target is stronger, that is the cue for which \(p\) is higher. Unfortunately, we cannot measure \(p\). However, we can collect a noisy signal of this parameter using an auxiliary task. Concretely, we use reaction time in a 2-AFC task in which participants are presented with a word and must select the matching image.

To map this measure onto the model’s \(p\) parameter, we take advantage of the fact that the expected time to reach threshold in the model is \(E[t] = \theta / p\) which in implies that \(\log E[t] = \log(\theta) - \log(p)\). This suggests that, in broad strokes, the \(p\) parameter should be log-linearly related to response times. The exact nature of this relationship is unclear, however, given that the 2-AFC task is quite different from a cued recall task. Thus, we simply normalize (Z-score) the \(\log(p)\) parameter and the 2AFC-RT measure (the latter within subject) to put them on roughtly the same scale. Finally, to account for the fact that the 2AFC measure is very noisy, we corrupt \(\log(p)\) with \(\sigma=3\) Gaussian noise.

In the simulations, we sample the parameter \(p\) from a \(\text{Beta}(2, 6)\) distribution, which was chosen because it makes some of the plots look better.

Fixation time course

The key model prediction is that the probability of looking at the cue with greater memory strength increases over time.

normalized_timestep = function(long) {
    long %>% 
        group_by(trial) %>%
        mutate(prop_duration = duration / sum(duration)) %>% 
        ungroup() %>% 
        mutate(n_step=round(prop_duration * 100)) %>% 
        uncount(n_step) %>% 
        group_by(trial) %>% 
        mutate(normalized_timestep = row_number())
}

long %>% 
    normalized_timestep %>% 
    drop_na(strength_diff) %>% 
    ggplot(aes(normalized_timestep/100, fix_stronger, group = strength_diff, color=strength_diff)) +
    geom_smooth(se=F) + 
    ylim(0, 1) +
    facet_grid(~name) +
    labs(x="Normalized Time", y="Probability Fixate\nStronger Cue", color="Strength Difference") +
    geom_hline(yintercept=0.5) +
    theme(legend.position="top")

In both the optimal model simulations and the human data, the probability of fixating the stronger cue steadily increases over the course of the trial, and this increase is more pronounced when there is a larger difference in strength.

In the random model, we only see a slight spike at the end of the trial. This spike is due to the fact that the model always remembers the item it looks at last, and it is also more likely to remember the item with a stronger memory.

The random commitment model does not show this effect because the item it fixates last (and thus remembers) is usually determined by the random commitment.

Individual fixation durations

Although the random models we designed did not demonstrate the smoothly increasing probability shown by both humans and the optimal model, it is possible that the aforementioned last-fixation phenomenon could be partially driving that result. We can avoid this confound by looking at the duration of individual fixations, conditioning on the fact that each is not the final fixation. The model predicts that fixations should be longer for stronger cues, and (for non-initial fixations) when the competing cue is weaker.

For each of the first three fixations, we will plot model predictions alongside human data. Because we are excluding final fixations, different participants are better represented in some part of the x axis than others. This produces Simpon’s-paradox-like effects. Thus, we plot the fixed effect prediction of a mixed effects model rather than a generic fixed effects model. The printed regression tables describe the fixed effects of the plotted random effects model applied to the human data.

df %>% 
    filter(n_pres >= 2) %>% 
    regress(strength_first, first_pres_time) # plot

df %>% 
    filter(n_pres >= 3) %>% 
    regress(rel_strength, second_pres_time) # plot

df %>% 
    filter(n_pres >= 4) %>% 
    regress(rel_strength, third_pres_time)

Overall fixation proportion

The simplest test of rational memory: do people spend more time looking at the cue that they have a stronger memory of? This analysis only considers trials where both cues are seen.

df %>% 
    filter(n_pres >= 2) %>% 
    regress(rel_strength, prop_first)

This is suggestive, but there’s a caveat.

df %>% 
    filter(n_pres >= 2) %>% 
    regress_interaction(rel_strength, last_pres, prop_first)

df %>% 
    mutate(last_pres_first = as.numeric(last_pres == "first")) %>% 
    regress(rel_strength, last_pres_first)

long %>% 
    # filter(name == "Human") %>% 
    ggplot(aes(last_fix==1, duration)) + 
    stat_summary(fun.data=mean_cl_boot, geom="bar", fill="white", color="black") +
    stat_summary(fun.data=mean_cl_boot, geom="errorbar", width=0.2) +
    facet_grid(~name) + 
    scale_x_discrete(name="Fixation Type", labels=c("Non-final", "Final")) +
    ylab("Duration")

last_diff = long %>% 
    filter(name == "Human") %>% 
    with(tapply(duration, last_fix, mean)) %>% 
    diff

Probability of remembering first word

df %>%
    filter(response_type == "correct") %>% 
    mutate(choose_first = as.numeric(choose_first)) %>% 
    regress(rel_strength, choose_first) +
    ylab("Prob Select First Cue")

Reaction time by chosen strength

df %>% 
    filter(response_type == "correct") %>% 
    regress(chosen_strength, rt)

Last fixation duration by strength

df %>% 
    filter(response_type == "correct") %>% 
    filter(n_pres > 0) %>% 
    mutate(
        last_pres_time = map_dbl(presentation_times, last),
        last_rel_strength = if_else(last_pres == "first", rel_strength, 1 - rel_strength),
        last_strength = if_else(last_pres == "first", strength_first, strength_second)
    ) %>% 
    regress(last_strength, last_pres_time)

Timeout rate

df %>% 
    group_by(name) %>% 
    summarise(timeout_rate = mean(response_type == "timeout")) %>% 
    kable(digits=2)

Reaction time

df %>% 
    ggplot(aes(rt)) +
    geom_density() +
    facet_grid(~name) +
    labs(x="Reaction Time", y="Density")

Number of fixations

df %>% 
    filter(n_pres < 10) %>% 
    ggplot(aes(n_pres, ..prop..)) +
    geom_bar() +
    facet_grid(~name) +
    labs(x="Number of Fixations", y="Proportion of Trials")